Three Essays on Structural Stability of Time Series Models

(1)

Three Essays on Structural Stability

of Time Series Models

Inauguraldissertation

zur

Erlangung des Doktorgrades

der

Wirtschafts- und Sozialwissenschaftlichen Fakult¨

at

der

Universit¨

at zu K¨

oln

2019

vorgelegt

von

Sven Otto

aus

Hachenburg

(2)

Referent: Prof. Dr. J¨org Breitung

Koreferent: Prof. Dr. Dominik Wied

(3)

Acknowledgements

This thesis would not have been possible without the support of many people, and I owe them a great amount of gratitude. First and foremost, I would like to thank my main supervisor J¨org Breitung for his guidance, encouragement, and steady support. During my time as a research assistant at his chair I learned a lot from his way of doing research, and his approaches of finding simple and elegant solutions to complex econometric problems inspired and motivated me. He was always willing and interested to discuss my projects and gave me invaluable comments and advice.

I am thankful to Dominik Wied for his support and feedback concerning technical issues and for accepting to be my second supervisor. Furthermore, I would like to thank Matei Demetrescu for his very helpful comments and support regarding the first chapter of this work, as well as Robinson Kruse-Becher for his support on my academic path. The joint project with Nazarii Salish was a great experience and I would like to thank him for many hours of fruitful discussions during my research visit in Madrid and via video chat. The last five years wouldn’t have been the same without my colleagues and fellow doctoral students at the Institute of Econometrics and Statistics at the University of Cologne, and I am thankful for many interesting academic and non-academic discussions, the pleasant working atmosphere, and the wonderful time.

Most importantly, I am deeply grateful to my family for their unconditional support over the years. Nora deserves more than gratitude for her support and belief in me.

(4)

List of Figures

1.1 Plots of the trend functions . . . 17

2.1 Retrospective testing and monitoring . . . 48

2.2 Illustrative example for the backward CUSUM with a break in the mean . 57 2.3 Asymptotic local power curves for retrospective testing . . . 61

2.4 Asymptotic local mean delay curves for monitoring with m = 4 . . . 63

2.5 Size distributions of the retrospective detectors . . . 63

2.6 Size distributions of the monitoring detectors with m = 10 . . . 63

3.1 Nelson-Siegel loading functions . . . 80

3.2 Yields of U.S. Treasury bonds . . . 81

3.3 Fitted Nelson-Siegel curve and cubic B-spline representation . . . 82

3.4 Explained variance of the factors and scree plot . . . 91

3.5 Empirical functional principal components . . . 92

3.6 Effects of the first six functional principal components . . . 93

3.7 Empirical functional principal component scores . . . 94

(7)

List of Tables

1.1 Asymptotic critical values for the fixed-b test . . . 11

1.2 Trend functions . . . 16

1.3 Size and size-adjusted powers under the zero-trend specification . . . 19

1.4 Size and size-adjusted powers under different trends and i.i.d. errors (1/2) . 20 1.5 Size and size-adjusted powers under different trends and i.i.d. errors (2/2) . 21 1.6 Size and size-adjusted powers under different trends and AR(1) errors . . . 22

1.7 Size and size-adjusted powers of robust tests under constant trend and variance . . . 23

1.8 Size and size-adjusted power of robust tests under breaks in trend and variance . . . 24

1.9 Unit root tests applied to inflation rates . . . 26

2.1 Asymptotic critical values for the retrospective tests . . . 64

2.2 Empirical sizes of the retrospective tests . . . 64

2.3 Size-adjusted powers of the retrospective tests . . . 65

2.4 Asymptotic critical values for the stacked backward CUSUM monitoring . 66 2.5 Empirical sizes of the infinite horizon monitoring detectors . . . 67

2.6 Empirical mean detection delays of the monitoring detectors . . . 67

3.1 Root mean square forecast errors . . . 96

3.2 Diebold-Mariano test statistics . . . 97

3.3 Coverage rates and average widths of one-month-ahead interval forecasts . 98 3.4 Coverage rates of one-month-ahead simultaneous prediction bands . . . 98

(8)

Introduction

The analysis of the inherent structure of time-dependent data is crucial for the selection of a suitable model. This thesis is comprised of three self-contained essays on time series models and the identification of their stability properties. Hypothesis tests are particularly useful in this context. While the first chapter presents a unit root test that is robust against an unknown nonparametric trend, the second chapter deals with testing for structural change in linear regression models. The third chapter is devoted to the analysis of the functional dependence structure of bond yields with different maturities, and discusses the identification and estimation of a functional factor model for yield curves from the perspective of functional data analysis. A more detailed description of each chapter is given in the remainder of the introduction.

While the first chapter is based on a single-author paper (see Otto 2019), the latter two chapters are joint works with J¨org Breitung (see Otto and Breitung 2019) and Nazarii Salish (see Otto and Salish 2019) respectively.

Chapter 1 The literature on unit root testing is large and comprehensive, beginning with the seminal works of Dickey and Fuller (1979), Said and Dickey (1984), Phillips (1987), and Phillips and Perron (1988). Elliott et al. (1996) presented a unit root test that exhibits optimality properties. These conventional unit root tests include the assump-tion that the deterministic component is either constant or linear. Since a misspecified trend model leads to power losses, many studies focused on unit root testing under a more flexible parametric structure for the trend component, such as structural break and smooth transition models with unknown breakpoint and magnitude, and approximations by Chebyshev polynomials and Fourier series. However, little attention has been devoted to the nonparametric treatment of deterministic trends.

The testing approach presented in this chapter is based on the idea that in a window every smooth trend can be approximated well by a constant if the window size is small enough. The time series is proposed to be divided into overlapping blocks, and the unit root hypothesis is tested based on a pooled regression across all these blocks. Since the

(9)

trend is estimated for each block separately, the pooled OLS estimator filters out the trend component, asymptotically. Both fixed-b and small-b block asymptotics are considered, and the limiting distributions of the t-statistics for the unit root hypothesis are derived. A nuisance parameter correction under heteroskedasticity provides heteroskedasticity-robust tests, and serial correlation is accounted for by a pre-whitening scheme. Furthermore, the limiting distributions under local alternatives are derived. An extensive Monte Carlo study shows that the proposed tests yield improved power results under both slowly varying trends and sharp breaks when compared to conventional unit root tests. Moreover, the tests are well sized and comparable to the conventional tests in terms of power if the trend is constant.

Chapter 2 The CUSUM test of Brown et al. (1975) for detecting changes in the coef-ficients of a linear regression and the corresponding monitoring procedure of Chu et al. (1996) suffer from low power and large detection delay. Therefore, two alternative detec-tor statistics are proposed. The backward CUSUM detecdetec-tor sequentially cumulates the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector considers a triangular array of backward cumulated residuals. While both the backward CUSUM detector and the stacked backward CUSUM detector are suitable for retrospective testing, only the stacked backward CUSUM detector can be monitored on-line. The limiting distributions of the maximum statistics under suitable sequences of alternatives are derived for retrospective testing and fixed endpoint monitoring. In the retrospective testing context, the local power of the tests is shown to be substantially higher than that for the conventional CUSUM test if a single break occurs after one third of the sample size. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is shown to be much shorter than that of the conventional monitor-ing CUSUM procedure. Moreover, an infinite horizon monitormonitor-ing procedure and critical values are presented.

Chapter 3 The problem of yield curve modeling and forecasting from a functional time series perspective is discussed. In the fashion of the vector-valued factor models of Stock and Watson (2002) and Bai (2003), a functional factor model for yield curves is studied, in which the factors follow some linear autoregressive process. The model is identified by imposing suitable orthogonality conditions on the factors and the loading functions, while both factors and loadings are unknown. The model can be seen as an extension of the yield curve models by Nelson and Siegel (1987) and Diebold and Li (2006), in which the loadings are fixed and predefined. By applying the least squares principle, a functional principal

(10)

components based estimator is obtained, which is shown to be consistent. Furthermore, the minimum mean squared error forecast from the dynamic functional factor model is derived. By imposing normality of the factors and the errors, pointwise and simultaneous prediction bands are obtained from the forecast error curve distribution. The accuracy of the predictions and prediction bands is discussed in an out-of-sample experiment with monthly yield curves of U.S. Treasuries.

(11)

Chapter 1 Unit Root Testing with Slowly

Varying Trends

1.1 Introduction

It is widely debated in the time series literature whether macroeconomic variables such as GDP, inflation, and interest rates are I(1) or I(0) around a deterministic trend. Dickey-Fuller-type unit root tests often fail to reject the null hypothesis for these time series. The trend component of a time series yt is typically treated as known up to some parameter vector. The most commonly applied unit root tests, such as those developed by Dickey and Fuller (1979), Said and Dickey (1984), Phillips (1987), Phillips and Perron (1988), and Elliott et al. (1996), impose either a constant or a linear trend model. If, however, the deterministic trend component is nonlinear, highly persistent trend-stationary processes can be hardly distinguishable from unit root processes.

It is not only a misspecified trend model that may lead to high power losses, as an overparameterized model can also reduce the power of unit root tests. Therefore, many authors have suggested applying trend models that seem more suitable for macro data. Broken trend models with one-time changes in mean or slope with known breakpoint were first studied by Perron (1989) and Rappoport and Reichlin (1989). Christiano (1992) demonstrated that a broken trend model with an unknown breakpoint is more adequate, and Zivot and Andrews (1992), as well as Banerjee et al. (1992), proposed unit root tests for this framework. Structural changes in innovation variances were studied by Hamori and Tokihisa (1997), Kim et al. (2002), and Cavaliere (2005), while Cavaliere et al. (2011) considered unit root testing under broken trends together with nonstationary volatility. Leybourne et al. (1998), Kapetanios et al. (2003), and Kılı¸c (2011) allowed

(12)

for exponential smooth transitions from one trend regime to another. Bierens (1997) approximated a nonlinear mean function with Chebyshev polynomials, and Enders and Lee (2012) proposed a Fourier series approximation of the trend, which are approaches that can be used when the exact form and date of structural changes are unknown.

Dickey-Fuller-type tests are based on the t-statistic of the first-order autoregressive parameter. In case of a constant trend, the estimator is derived from a regression of ∆yt on (yt−1− y), where y is the sample mean. Schmidt and Phillips (1992) estimated the constant by the initial observation, which results in a regression of ∆yt on (yt−1 − y1). Whereas a constant is often not a good approximation, in a small block, a moderately varying trend can be approximated quite closely by a constant. To exploit this fact, we divide the series into T − B overlapping blocks of length B. As the blocks can be considered as units of a panel, we follow the panel unit root tests proposed by Breitung (2000) and Levin et al. (2002) and consider a pooled regression of ∆yj+t on (yj+t−1− yj) for 2 ≤ t ≤ T and 1 ≤ j ≤ T − B. The deterministic function is approximated locally by a constant. Under a general class of piecewise continuous nonparametric trend functions, the resulting pooled estimator is consistent as B, T → ∞. The limiting null distribution of the t-statistic is a functional of Brownian motions under fixed-b asymptotics. Under small-b asymptotics, a normal distribution is obtained.

The chapter is organized as follows: In Section 1.2 the autoregressive model with independent and homoskedastic errors is analyzed together with the asymptotic behavior of the pooled least squares estimator in the presence of a general nonparametric trend model. For both fixed-b and small-b block asymptotics, the limiting distributions are derived under both the unit root hypothesis and under local alternatives. Section 1.3 considers pseudo t-statistics for unit root testing, while Section 1.4 demonstrates that, under heteroskedasticity, nuisance parameters appear in the limiting distributions. The estimation of these parameters is discussed, and heteroskedasticity-robust test statistics are provided. In Section 1.5, a pre-whitening procedure is proposed in order to account for short-run dynamics, while Section 1.6 reports on Monte Carlo simulations. The tests are found to have only minor size distortions in small samples and are sized correctly in larger samples. It is shown that in the presence of slowly varying trends, pooled tests tend to yield higher power than conventional unit root tests. In Section 1.7, these tests are applied to the monthly inflation rates of 25 countries. The results provide some evidence in favor of inflation rates being trend-stationary around a slowly varying deterministic component. Finally, Section 1.8 presents the conclusion.

(13)

1.2 The pooled estimator

We are interested in inference concerning the autoregressive parameter ρ in the model yt= dt+ xt, xt = ρxt−1+ ut, t = 1, . . . , T,

where ρ is close or equal to one. The deterministic trend component dt is treated as nonstochastic and fixed in repeated samples, where its functional form is nonparametric and unknown.

Assumption 1.1 (trend component). The trend component is given by dt = d(t/T ), where d(r) is a piecewise Lipschitz continuous function.

Note that any continuously differentiable function is Lipschitz continuous. Lipschitz functions are locally close to a constant value in the sense that there exists some C < ∞ such that |d(r) − d(s)| ≤ C|r − s| for all r, s ∈ R. The piecewise Lipschitz condition allows for a partition with a finite number of intervals, such that d(r) is Lipschitz continuous on each interval. This includes both smooth changes as well as abrupt breaks in the trend function. For the initial value, it is assumed that E[x2

0] < ∞. We introduce the pooled estimator and the unit root test statistics under the following simplified assumptions on the error term, which are relaxed in the subsequent sections:

Assumption 1.2 (i.i.d. errors). The process {ut}t∈N is independently distributed, where E[ut] = 0, E[u2t] = σ2 and E[u4t] < ∞.

The principal approach to dealing with a nonparametric, slowly varying trend is to approximate the unknown trend locally by a constant. Let B be some blocklength that satisfies 2 ≤ B < T . We divide the time series into T − B overlapping blocks of length B and then block-wise estimate ρ via OLS under a constant trend specification. In the fashion of Schmidt and Phillips (1992), as well as Breitung and Meyer (1994), the constant trend is approximated by the first observation in each block. Thereafter, by pooling the T − B individual block regressions, we obtain the following regression equation:

∆yt+j = φ(yt+j−1− yj) + ut+j, t = 2, . . . , B, j = 1, . . . , T − B, where φ = ρ − 1. The pooled OLS estimator is formulated as

ˆ φ = ˆρ − 1 = PT −B j=1 PB t=2∆yt+j(yt+j−1− yj) PT −B j=1 PB t=2(yt+j−1− yj)2 .

(14)

In the following, we derive the asymptotic properties for the numerator and the denomi-nator separately. The numerator and denomidenomi-nator statistics are defined as

Y1,T = 1 B3/2_T1/2 T −B X j=1 B X t=2 ∆yt+j(yt+j−1− yj), Y2,T = 1 B2_T T −B X j=1 B X t=2 (yt+j−1− yj)2,

such that √BT ( ˆρ − 1) = Y1,T/Y2,T. Their counterparts for a zero trend component are given by X1,T = 1 B3/2_T1/2 T −B X j=1 B X t=2 ∆xt+j(xt+j−1− xj), X2,T = 1 B2_T T −B X j=1 B X t=2 (xt+j−1− xj)2.

In what follows, we show that, under the block procedure, the trend component can be ignored asymptotically. All asymptotic results are jointly derived for B, T → ∞. While the statistics X1,T and X2,T are infeasible if dt is unknown, they can be well approximated by Y1,T and Y2,T in the following sense:

Lemma 1.1. Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.2. Then, |Y1,T − X1,T| = oP(1), and |Y2,T − X2,T| = oP(1) as B, T → ∞.

The block procedure filters out the trend component in the numerator and the denom-inator asymptotically. Hence, applying Slutsky’s theorem, we can write

√ BT ( ˆρ − 1) = Y1,T Y2,T = X1,T X2,T + oP(1).

In order to obtain the limiting distribution, we formulate the following properties for the numerator and denominator statistics:

Lemma 1.2. Let ρ = 1 − c/√BT with c ≥ 0, and let ut satisfy Assumption 1.2. Then, as B, T → ∞, it follows that

(a) X1,T =PT_j=1qj,T − c · (σ2/2 + oP(1)), where {qj,T}j≤T,T ∈N is a martingale difference array with V ar T X j=1 qj,T = σ4(B − 1)((T − B)(2B − 1) − 2(B − 2)) 6B2_T , (b) E[X2,T] = σ2 (T − B)(B − 1) 2BT + c · O(B 1/2_T−1/2

(15)

Lemmas 1.1 and 1.2 imply that V ar[Y2,T] = O(1) if B = O(T ), whereas V ar[Y2,T] = o(1) if B = o(T ). This suggests distinguishing between these fundamentally different types of blocklength asymptotics. The fixed-b approach denotes the case where the relative blocklength B/T converges to some value b with 0 < b < 1, such that B and T grow at the same rate. In the small-b approach, we consider a relative blocklength that converges to zero, while B, T → ∞.1 As the blocks are overlapping, the error terms in the pooled regression equation are correlated, but, fortunately, the correlation structure is known by construction. Lemmas 1.1 and 1.2 imply that V ar[Y1,T] → σ4/3 and E[Y2,T] → σ2/2 as B/T → 0 and B, T → ∞. Together with the central limit theorem for martingale difference arrays, the following asymptotic result can be stated:

Theorem 1.1 (small-b asymptotics). Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.2. Let B/T → 0 as B, T → ∞. Then,

Y1,T D −→ N − cσ 2 2 , σ4 3 , and Y2,T p −→ σ 2 2 .

As a direct consequence, the pooled estimator is asymptotically normally distributed under small-b asymptotics. Together with Slutsky’s theorem, it follows that

√

BT ( ˆρ − 1)−→ N (0, 4/3)D

under the unit root hypothesis, which is given by ρ = 1 or equivalently by c = 0. Under fixed-b asymptotics, the numerator and denominator statistics can be represented as a partial sum process of the innovations. The functional central limit theorem then yields the following asymptotic result:

Theorem 1.2 (fixed-b asymptotics). Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.2. Let 0 < b < 1, and let B/T → b as B, T → ∞. Then Y1,T Y2,T ! D −→ σ2 2b3/2( R1−b 0 (Jc/b(b + r) − Jc/b(r)) 2_{− b(1 − b))} σ2 b2 R1−b 0 Rb+r r (Jc/b(s) − Jc/b(r)) 2_{ds dr} ! ,

where Jc(r) is an Ornstein-Uhlenbeck process.

1_{Note that the terminology “fixed-b and small-b asymptotics” has also been used in the context of} long-run variance estimation. Whereas Kiefer and Vogelsang (2005) use this wording for the asymptotics of the ratio of the truncation point to the sample size, we consider the ratio of the blocklength to the sample size.

(16)

Consequently, the pooled estimator is asymptotically represented as a functional of a standard Brownian motion W (r) under the unit root hypothesis. If ρ = 1, then Theorem 1.2, together with the continuous mapping theorem, implies that

√ BT ( ˆρ − 1)−→D b 1/2R1−b 0 (W (b + r) − W (r)) 2 dr + b3/2_{(1 − b)} 2R₀1−bR_rb+r(W (s) − W (r))2 ds dr

under fixed-b asymptotics. In comparison to the limiting distribution of the ρ-statistic in the Dickey-Fuller framework, the functional includes an additional integral, which results from pooling the block regressions.

1.3 Pseudo t-statistics for unit root testing

The principal concept of Dickey-Fuller-type unit root tests is to consider a t-test for the null hypothesis H0 : ρ = 1. Following this approach in the pooled regression framework, the numerator of the t-statistic can be represented as ˆρ − 1 = Y1,T(Y2,T

√

BT )−1. The standard error is obtained from the conditional variance of ˆρ. Let

s2_ρ_ˆ= ˆσ2 T −B_X j=1 B X t=2 (yt+j−1− yj)2 −1 = σˆ 2 Y2,TB2T ,

where ˆσ2 _{denotes some consistent estimator for the error variance σ}2_{. The conventional} t-statistic is then represented as ( ˆρ−1)/sρˆ=

√

BY1,T/p ˆσ2Y2,T and diverges in probability under H0. Accordingly, we consider a pseudo t-statistic of the form

τ = ρ − 1ˆ sρˆ √ B = Y1,T ˆ σpY2,T ,

which is OP(1) under both small-b and fixed-b asymptotics. We consider the residuals ˆ

ut = yt− ˆρyt−1 for t = 2, . . . , T and their sample mean û = T−1PT_j=1uˆj. For the error variance estimation, we distinguish between fixed-b and small-b block asymptotics and define ˆ σ_sb2 = PT −B j=1 PB t=1 ˆ uj+t− _B1 PB_k=1uˆj+k 2 (T − B)(B − 1) , σˆ 2 fb = 1 T T X j=2 (ûj− û)2.

(17)

Lemma 1.3. Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.2.

(a) Let B/T → 0 as B, T → ∞. Then, ˆσ2_sb−→ σp 2_.

(b) Let 0 < b < 1, and let B/T → b as B, T → ∞. Then, ˆσ2 fb

p −→ σ2_.

In what follows, the pseudo t-tests are defined. For the small-b pseudo t-statistic, we scale Y1,T and Y2,T by their finite sample variance and their expectation from Lemma 1.2 in order to avoid small-sample size distortions. Let

v_T2 = (T − B)(2B − 1) − 2(B − 2)

3B(T − B) ,

which is equal to σ−2V ar[X1,T]/E[X2,T] under the unit root hypothesis. The small-b pseudo t-statistic is then defined as

τ -SB = Y1,T vTσˆsbpY2,T = PT −B j=1 PB t=2∆yt+j(yt+j−1− yj) ˆ σsb q (T −B)(2B−1)−2(B−2) 3(T −B) PT −B j=1 PB t=2(yt+j−1− yj)2 .

For the fixed-b statistic, we consider the unscaled versions and define

τ -FB = Y1,T ˆ σfbpY2,T = PT −B j=1 PB t=2∆yt+j(yt+j−1− yj) ˆ σfb q BPT −B j=1 PB t=2(yt+j−1− yj)2 .

The unit root hypothesis is rejected in favor of stationarity if the test statistic is smaller than the α-quantile of the limiting distribution under H0, where α is the significance level. From Theorems 1.1 and 1.2 and Lemma 1.3, together with the continuous mapping theorem and Slutsky’s theorem, the following limiting result can be stated:

Corollary 1.1. Let ρ = 1, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.2.

(a) Let B/T → 0 as B, T → ∞. Then, τ -SB−→ N (0, 1).D (b) Let 0 < b < 1, and let B/T → b as B, T → ∞. Then,

τ -FB−→D R1−b 0 (W (b + r) − W (r)) 2 dr − b(1 − b) 2 q bR₀1−bR_rb+r(W (s) − W (r))2 ds dr ,

(18)

Table 1.1: Asymptotic critical values for the fixed-b test α B/T 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 -0.788 -0.812 -0.815 -0.799 -0.761 -0.701 -0.623 -0.520 -0.377 0.1 -1.126 -1.128 -1.104 -1.055 -0.987 -0.903 -0.798 -0.664 -0.486 0.05 -1.403 -1.375 -1.327 -1.257 -1.169 -1.067 -0.939 -0.781 -0.573 0.04 -1.486 -1.446 -1.391 -1.318 -1.222 -1.113 -0.978 -0.814 -0.600 0.03 -1.582 -1.534 -1.471 -1.394 -1.291 -1.169 -1.025 -0.855 -0.630 0.02 -1.709 -1.650 -1.579 -1.489 -1.374 -1.246 -1.094 -0.909 -0.669 0.01 -1.904 -1.830 -1.745 -1.639 -1.511 -1.361 -1.191 -0.995 -0.729 0.001 -2.431 -2.320 -2.203 -2.042 -1.882 -1.692 -1.480 -1.226 -0.905

Note: The sample paths of the standard Brownian motions contained in the asymptotic null distribution of τ -FB are simulated by a discretized version of W (r) on a grid of 50,000 equidistant points. The empirical quantiles are obtained from 100,000 Monte Carlo repetitions.

For τ -SB we can rely on standard normal quantiles as critical values. However, the limiting distribution of τ -FB is nonstandard. Table 1.1 presents simulated left-tailed quantiles of the null distribution for various relative blocklengths B/T and significance levels.

From Theorems 1.1 and 1.2, it follows that the tests have power against alternatives of the form ρ = 1 − c/√BT , where c > 0.

1.4 Testing under heteroskedasticity

While stationary time-varying conditional variances such as ARCH and GARCH pro-cesses do not affect unit root testing, Hamori and Tokihisa (1997) showed that per-manent changes in volatility, in contrast, dramatically alter the limiting distributions of unit root tests. Kim et al. (2002) reported that an abrupt break in the innovation variance can produce spurious rejections, while Cavaliere (2005) showed that nonsta-tionary volatility induces a time-shift in the right-hand-side process of the functional central limit theorem. A variance-transformed Brownian process Wη(r) appears in the limiting distributions of Dickey-Fuller-type unit root tests. Given the variance profile η(s) = (R₀1σ(r)2dr)−1R₀sσ(r)2dr, the transformed process is defined as Wη(r) = W (η(r)), where 0 ≤ r ≤ 1. In what follows, we relax Assumption 1.2 and allow for heteroskedastic errors.

Assumption 1.3 (heteroskedastic errors). The process {ut}t∈N is independently distrib-uted with E[ut] = 0, E[u2t] = σ2t and E[u4t] < ∞, where σt= σ(t/T ). The function σ(r) is c`adl`ag, non-stochastic, strictly positive, and bounded.

(19)

and can be formulated under heteroskedasticity as follows:

Lemma 1.4. Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.3. Then, |Y1,T − X1,T| = oP(1), and |Y2,T − X2,T| = oP(1) as B, T → ∞.

However, nuisance parameters then appear in the limiting distributions of the numer-ator and denominnumer-ator statistics.

Theorem 1.3. Let ρ = 1 − c/√BT with c ≥ 0, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.3.

(a) Let B/T → 0 as B, T → ∞. Then,

Y1,T D −→ N − c 2 Z 1 0 σ2(r) dr, 1 3 Z 1 0 σ4(r) dr , and Y2,T p −→ 1 2 Z 1 0 σ2(r) dr.

(b) Let 0 < b < 1, and let B/T → b as B, T → ∞. Then, Y1,T Y2,T ! D −→ R1 0 σ2(r) dr 2b3/2 ( R1−b 0 (Jc,b,η(b + r) − Jc,b,η(r)) 2 _{− b(1 − b))} R1 0 σ2(r) dr b2 R1−b 0 Rb+r r (Jc,b,η(s) − Jc,b,η(r)) 2_{ds dr} ! ,

where Jc,b,η(r) is a variance-transformed Ornstein-Uhlenbeck process, which is defined

as Jc,b,η(r) =

Rr

0 e

−(r−s)c/b_dW

η(s).

In order to correct for the additional nuisance terms, we consider the following esti-mators. Let ˆ κ2 = PT −B j=1 PB t=1 uˆj− û 2 ˆ uj+t− _B1 PB_k=1uˆj+k 2 (T − B)(B − 1) and let ˆ η(s) = PbsT c j=1 ˆ uj − _{bsT c}1 P bsT c k=1 uˆk 2 + (sT − bsT c) ˆ ubsT c+1− _{bsT c+1}1 P bsT c+1 k=1 uˆk 2 PT j=1(ûj− û)2 ,

where s ∈ [0, 1]. The robust small-b statistic is then defined as

τ -SBH = Y1,T vTκˆˆσ−1_sbpY2,T = PT −B j=1 PB t=2∆yt+j(yt+j−1− yj) ˆ κˆσ_sb−1q(T −B)(2B−1)−2(B−2)_{3(T −B)} PT −B j=1 PB t=2(yt+j−1− yj)2 .

(20)

Under fixed-b asymptotics, the nuisance term appears in the Gaussian process itself. By means of transforming the data with its inverse variance profile, Cavaliere and Taylor (2007) showed that the time-transformation in the Gaussian limiting processes can be inverted. Accordingly, we fix some auxiliary sample size eT ≥ T and consider the time-transformed series ˜yt = ybˆη−1_{(t/ e}_{T ) e}_{T c} for t = 1, . . . , eT , where ˆη−1(s) is the inverse function

of ˆη(s). In practice, the observed time series is transformed in such a way that copies of adjacent observations between the sample points are inserted in highly volatilie periods. Since eT can be set arbitrarily high, we do not need to discard any observations. We replace the original series in the test statistic by the time-transformed series and define

e Y1,T = 1 B3/2 e T1/2 e T −B X j=1 B X t=2 ∆˜yt+j(˜yt+j−1− ˜yj), Ye2,T = 1 B2 e T e T −B X j=1 B X t=2 (˜yt+j−1− ˜yj)2,

which yields the fixed-b heteroskedasticity-robust statistic

τ -FBH = Ye1,T ˆ σfb q e Y2,T = PT −Be j=1 PB t=2∆˜yt+j(˜yt+j−1− ˜yj) ˆ σfb q BPT −Be j=1 PB t=2(˜yt+j−1− ˜yj)2 .

Theorem 1.4. Let ρ = 1, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.3.

(a) Let B/T → 0 as B, T → ∞. Then, ˆσ2_sb −→p R1

0 σ

2_{(r) dr, ˆ}_κ2 _−→p R1

0 σ

4_{(r) dr, and} τ -SBH−→ N (0, 1).D

(b) Let 0 < b < 1, and let B/ eT → b as B, eT → ∞. Then, ˆσ2 fb p −→ R1 0 σ 2_{(r) dr,} ˆ

η(s)−→ η(s) uniformly for all s ∈ [0, 1], andp

τ -FBH −→D R1−b 0 (W (b + r) − W (r)) 2 dr + b(1 − b) 2 q bR1−b 0 Rb+r r (W (s) − W (r)) 2 ds dr .

The limiting distributions under the unit root hypothesis of the heteroskedasticity-robust test statistics coincide with those obtained in Section 1.3 under homoskedasticity. Hence, the critical values from those tests can be retained. For τ -SBH, we consider stan-dard normal quantiles, and, for τ -FBH, we can apply the values from Table 1.1.

(21)

1.5 Testing under short-run dynamics

A more realistic scenario for macroeconomic variables is that error terms are serially correlated. We impose the following assumption on the error process:

Assumption 1.4 (serially correlated errors). The process {ut}t∈N possesses the station-ary AR(p) representation ut = Pp_i=1θiut−i + t. The process {t}t∈Z is independently distributed with E[t] = 0, E[2t] = σt2 and E[4t] < ∞, where σt = σ(t/T ). The function σ(r) is c`adl`ag, non-stochastic, strictly positive, and bounded. The lag order p satisfies T−1/4p → 0.

In the fashion of Said and Dickey (1984), we allow the lag order to grow with the sample size. Asymptotically, this allows for fairly general forms of serially correlated errors, such as stationary and invertible ARMA processes. In order to correct for the effect of short-run dynamics, we follow Breitung and Das (2005) and consider the pre-whitened series y∗_t = yt−Pp_i=1θiyt−i. The series decomposes into yt∗ = d∗t + x∗t, where the deterministic and the stochastic parts are given by d∗_t = dt −Pp_i=1θidt−i and x∗t = xt−Pp_i=1θixt−i respectively. Note that x∗_t − ρx∗

t−1 = t, where t satisfies the same conditions as ut under Assumption 1.3. Consequently, if the unit root statistics are defined in terms of

X_1,T∗ = 1 B3/2_T1/2 T −B X j=1 B X t=2 ∆x∗_t+j(x∗_t+j−1− x∗_j), X_2,T∗ = 1 B2_T T −B X j=1 B X t=2 (x∗_t+j−1− x∗_j)2

instead of X1,T and X2,T, their limiting distributions then coincide with those presented in the previous sections.

Since the autoregressive parameters of the error process are unknown, they need to be estimated. We augment the regression equation with lagged values of the differenced series, such that

∆yt= ϕyt−1+ p X

i=1

βi∆yt−i+ et, (1.1)

for t = p + 1, . . . , T , where et is a mean-zero error term. Let ˆϕ and ˆβ1, . . . , ˆβp denote the OLS estimators of the parameters. In the following, we show that ( ˆβ1, . . . , ˆβp)0 is consistent for (θ1, . . . , θp)0 under the unit root hypothesis:

Lemma 1.5. Let ρ = 1, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.4. Then max1≤i≤pp| ˆβi− θi| = oP(1) as B, T → ∞.

(22)

The estimated pre-whitened series is defined as ˆy∗_t = yt−Pp_i=1βˆiyt−i, and the corre-sponding numerator and denominator statistics are given by

ˆ Y_1,T∗ = 1 B3/2_T1/2 T −B X j=1 B X t=2 ∆ˆy_t+j∗ (ˆy_t+j−1∗ − ˆy_j∗), Yˆ_2,T∗ = 1 B2_T T −B X j=1 B X t=2 (ˆy∗_t+j−1− ˆy_j∗)2.

Lemma 1.6. Let ρ = 1, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.4. Then, | ˆY∗

1,T − X1,T∗ | = oP(1), and | ˆY2,T∗ − X2,T∗ | = oP(1) as B, T → ∞.

The estimators ˆσ_sb∗2, ˆσ_fb∗2, ˆκ∗2, and ˆη∗(s) are defined as their counterparts in Sections 1.3 and 1.4, except that the residuals are now defined as ˆut= ˆy∗t− ˆρ

∗ ˆ y_t−1∗ , where ˆρ∗ is given by √ BT ( ˆρ∗− 1) = ˆY∗ 1,T/ ˆY ∗

2,T. Analogously to Section 1.4, we consider the time-transformed pre-whitened series ˜y_t∗ = ˆy∗

bˆη−1_{(t/ e}_{T ) e}_{T c} for t = 1, . . . , eT , where eT ≥ T , and we define

e Y_1,T∗ = 1 B3/2 e T1/2 e T −B X j=1 B X t=2 ∆˜y_t+j∗ (˜y_t+j−1∗ − ˜y_j∗), Ye_2,T∗ = 1 B2 e T e T −B X j=1 B X t=2 (˜y∗_t+j−1− ˜y_j∗)2.

The pre-whitened versions of the test statistics are then given by

τ -SBPW= ˆ Y∗ 1,T vTσˆ_sb∗ q ˆ Y∗ 2,T , τ -SBH-PW = ˆ Y∗ 1,T vTκˆ∗ˆσ∗−1_sb q ˆ Y∗ 2,T , τ -FBPW = ˆ Y∗ 1,T ˆ σ_fb∗ q ˆ Y∗ 2,T , τ -FBH-PW = Ye ∗ 1,T ˆ σ∗ fb q e Y∗ 2,T .

Theorem 1.5. Let ρ = 1, let dt satisfy Assumption 1.1, and let ut satisfy Assumption 1.4.

(a) Let B/T → 0 as B, T → ∞. Then, ˆσ_sb∗2 −→p R1

0 σ 2_{(r) dr, ˆ}_κ∗2 _−→p R1 0 σ 4_{(r) dr, and} τ -SBH-PW−→ N (0, 1). Furthermore, τ -SBD PW _{−→ N (0, 1) if σ}D 2 t = σ2 for all t. (b) Let 0 < b < 1, and let B/ eT → b as B, eT → ∞. Then, ˆσ∗2_fb −→p R1

0 σ

2_{(r) dr,} ˆ

η(s)−→ η(s) uniformly for all s ∈ [0, 1], andp

τ -FBH-PW −→D R1−b 0 (W (b + r) − W (r)) 2 dr + b(1 − b) 2 q bR₀1−bR_rb+r(W (s) − W (r))2 ds dr .

(23)

Furthermore, if σ2

t = σ2 for all t ∈ N, then

τ -FBPW −→D R1−b 0 (W (b + r) − W (r)) 2 dr + b(1 − b) 2 q bR1−b 0 Rb+r r (W (s) − W (r)) 2 ds dr .

The lag order p is typically unknown in practice and can be chosen using conventional lag order selection methods, such as the Bayesian information criterion (BIC) or by the general-to-specific methodology in the fashion of Ng and Perron (1995). The maximum lag order is inspired by the rule of thumb provided by Schwert (1989) and is fixed as p∗ = b4 · (T /100)1/5_{c or as p}∗ _{= b12 · (T /100)}1/5_c.

1.6 Simulations

In this section, the finite sample performance of the unit root tests is evaluated by means of Monte Carlo simulations. The analysis includes several specifications for both the deterministic part dt and the stochastic part xt.

While the zero-trend dt = 0 is the main benchmark, we consider several other trends including sharp breaks and smooth changes of different shapes. The trend specifications are presented in Table 1.2 and Figure 1.1. The parameter λ determines the size of the break. Similar trend functions are also considered in Jones and Enders (2014) in order to evaluate the performance of the unit root test by Enders and Lee (2012).

Table 1.2: Trend functions

type of the trend functional form

1 sharp break d(r) = λ · 1_{r≤2/3}

2 u-shaped break d(r) = λ · 1_{r≤1/4}+ λ · 1_{r>3/4} 3 continuous break d(r) = λ · (4r · 1_{r>2/3}− 8/3)

4 u-shaped break in intercept d(r) = λ · (r1{r≤1/4}+ (r − 1)1{1/4<r≤3/4}+ r1{t>3/4})

5 LSTAR break d(r) = λ · (1 + exp(20(r − 0.75)))−1

6 offsetting LSTAR break d(r) = λ/(1 + exp(20(r − 0.2))) − 0.5λ/(1 + exp(20(r − 0.75))) 7 triangular break d(r) = λ · (2r1{r≤1/2}+ 2(1 − r)1{r>1/2})

8 Fourier break d(r) = λ · 0.5 cos(2πr)

Note: The functional form of the trend functions, which are considered in the Monte Carlo simulations, are presented. The parameter λ determines the size of the trend.

The stochastic part xt is simulated both under the null hypothesis ρ = 1 and the alternative hypothesis ρ = 0.9. For the errors ut, we consider an independent process as well as the AR(1) process ut = 0.5ut−1+twith standard normal innovations. Furthermore, results with heteroskedastic innovations using the variance function σ2(r) = λ · 1{r≤2/3}are presented. As noted by M¨uller and Elliott (2003), the power of a unit root test depends

(24)

Figure 1.1: Plots of the trend functions 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 sharp break r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 u−shaped break r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 continuous break r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 −3 −1 1 2 3

u−shaped break in intercept

r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 LSTAR break r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2

offsetting LSTAR breaks

r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 triangular break r d(r) 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 Fourier break r d(r)

Note: The plots of the of the trend functions from Table 1.2, which are considered in the Monte Carlo simulations, are presented. The trend size is λ = 3.

on the initial condition. Thus, we consider both the zero initial condition x0 = 0 as well as a random initial condition, where x0 = PT_k=1ρT −k˜k is simulated from i.i.d. standard normal innovations ˜k.

For the small-b tests, we consider blocklengths of the form B = Tγ _{with parameters} γ ∈ {0.5, 0.6, 0.7, 0.8}, and, for the fixed-b versions, we implement B = b · T with rela-tive blocklengths b ∈ {0.2, 0.3, 0.4, 0.5}. Size and power results are presented for τ -SB and τ -FB as well as for their pre-whitened and heteroskedasticity-robust versions. Both fixed lag augmentation as well as a flexible lag augmentation determined by the BIC are implemented. All empirical size levels are presented for a significance level of 5%, while the power results are size-adjusted. The models are simulated with 100,000 repetitions for sample sizes of T = 100 and T = 300.

In order to demonstrate the advantage of the fixed-b and small-b unit root tests, their finite sample results are compared to those obtained by conventional unit root tests. As the main benchmark, we consider the augmented Dickey-Fuller test by Said and Dickey (1984) with constant trend specification (ADF henceforth), which is the t-test for the hypothesis ϕ = 0 in the regression

∆yt = ϕyt−1+ β0+ p X

i=1

ξi∆yt−i+ et.

(25)

the deterministic trend function be given by the vector zt, and let c ∈ R. Furthermore, let yc,t = yt− cyt−1 and Zc,t = zt− czt−1 for t ≥ 2, and let yc,1 = y1 and Zc,1 = z1. The Dickey-Fuller GLS test is then the t-test for the hypothesis ϕ = 0 in the regression

∆y_td= ϕy_t−1d + p X i=1 ξi∆yt−id + et, where yd

t = yt− ˆβ0zt and where ˆβ is the OLS estimator from a regression of yc,t on Zc,t. For the constant trend specification (DF-GLS henceforth), let zt = 1 and c = 7, and, for the linear trend specification (DF-GLS-trend henceforth), zt = (1, t)0 and c = 13.5 are considered. Elliott et al. (1996) showed that the Dickey-Fuller GLS test is optimal for the zero initial condition x0 = 0.

An approach that does not assume a precise model for the trend component is that developed by Enders and Lee (2012) (EL henceforth). A flexible Fourier form is used to approximate smooth breaks in the trend function. Structural changes can be captured by the low frequency components of a series. In its simplest form, Enders and Lee (2012) considered the parametric trend model d(r) = α0+ γr + α1sin(2πr) + β1cos(2πr). More frequencies could be included, but doing so could lead to an over-fitting problem. The test works as follows: First, the auxiliary regression

∆yt= δ0+ δ1∆ sin(2πt/T ) + δ2∆ cos(2πt/T ) + vt

is considered with OLS estimates bδ0, bδ1, and bδ2. This yields the detrended series

e

St= yt− (y1− bδ0− bδ1sin(2π_T ) − bδ2cos(2π_T )) − bδ0t − (bδ1sin(2πt_T ) + bδ2cos(2πt_T )). Finally, the test statistic is given by the t-statistic for the null hypothesis ϕ = 0 in the regression

∆yt= ϕ eSt−1+ β0+ β1∆ sin(2πt/T ) + β2∆ cos(2πt/T ) + p X

i=1

ξi∆ eSt−i+ et.

For all tests, the lag augmentation order p is either fixed or flexibly determined by the BIC with a maximum lag order of p∗ = b4 · (T /100)1/5_c.

The results presented in Tables 1.3–1.5 indicate that the pooled tests are slightly undersized for smaller sample sizes, where the size distortions become larger as the break gets larger. However, for larger sample sizes, the size distortions decline. Overall, the size levels are similar to those obtained from using the conventional unit root tests.

(26)

Table 1.3: Size and size-adjusted powers under the zero-trend specification

zero initial condition random initial condition

T = 100 T = 300 T = 100 T = 300

ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9

i.i.d. errors – no lag augmentation (p=0)

τ -SB, B = T0.5 _0.057 _0.294 _0.056 _0.845 _0.056 _0.281 _0.058 _0.838 τ -SB, B = T0.6 _0.057 _0.351 _0.057 _0.952 _0.057 _0.334 _0.058 _0.948 τ -SB, B = T0.7 _0.054 _0.409 _0.056 _0.989 _0.052 _0.394 _0.057 _0.987 τ -SB, B = T0.8 0.040 0.445 0.045 0.997 0.039 0.420 0.046 0.996 τ -FB, B = 0.2T 0.049 0.390 0.049 0.991 0.047 0.378 0.051 0.989 τ -FB, B = 0.3T 0.051 0.428 0.049 0.996 0.050 0.410 0.050 0.995 τ -FB, B = 0.4T 0.053 0.443 0.050 0.997 0.051 0.421 0.051 0.996 τ -FB, B = 0.5T 0.053 0.452 0.050 0.997 0.052 0.422 0.051 0.997 ADF 0.054 0.310 0.052 0.995 0.053 0.334 0.052 0.996 DF-GLS 0.078 0.661 0.058 1.000 0.076 0.490 0.059 0.931 DF-GLS-trend 0.069 0.294 0.053 0.993 0.069 0.255 0.054 0.945 EL 0.061 0.117 0.054 0.761 0.061 0.113 0.054 0.736

AR(1) errors – fixed lag augmentation (p=1)

τ -SBPW_{, B = T}0.5 _0.010 _0.309 _0.020 _0.805 _0.008 _0.331 _0.018 _0.819 τ -SBPW_{, B = T}0.6 _0.021 _0.335 _0.036 _0.908 _0.017 _0.357 _0.033 _0.913 τ -SBPW_{, B = T}0.7 _0.031 _0.368 _0.044 _0.963 _0.027 _0.387 _0.043 _0.964 τ -SBPW_{, B = T}0.8 _0.026 _0.381 _0.040 _0.981 _0.021 _0.408 _0.038 _0.982 τ -FBPW, B = 0.2T 0.024 0.355 0.040 0.968 0.020 0.374 0.038 0.969 τ -FBPW, B = 0.3T 0.032 0.375 0.043 0.980 0.028 0.393 0.042 0.981 τ -FBPW_{, B = 0.4T} _0.037 _0.379 _0.045 _0.983 _0.031 _0.403 _0.044 _0.985 τ -FBPW_{, B = 0.5T} _0.039 _0.379 _0.046 _0.985 _0.033 _0.407 _0.044 _0.986 ADF 0.056 0.242 0.051 0.969 0.055 0.244 0.052 0.969 DF-GLS 0.077 0.589 0.058 1.000 0.079 0.522 0.059 0.990 DF-GLS-trend 0.071 0.239 0.052 0.968 0.069 0.229 0.053 0.951 EL 0.067 0.095 0.056 0.609 0.067 0.094 0.057 0.595

AR(1) errors – flexible lag augmentation where p is determined by BIC

τ -SBPW_{, B = T}0.5 _0.004 _0.357 _0.015 _0.837 _0.003 _0.397 _0.013 _0.850 τ -SBPW_{, B = T}0.6 _0.015 _0.366 _0.032 _0.913 _0.011 _0.395 _0.029 _0.921 τ -SBPW_{, B = T}0.7 _0.026 _0.387 _0.042 _0.960 _0.020 _0.414 _0.040 _0.962 τ -SBPW_{, B = T}0.8 _0.023 _0.394 _0.039 _0.978 _0.017 _0.425 _0.037 _0.979 τ -FBPW, B = 0.2T 0.019 0.377 0.039 0.964 0.014 0.406 0.036 0.967 τ -FBPW, B = 0.3T 0.028 0.389 0.043 0.976 0.023 0.415 0.040 0.978 τ -FBPW, B = 0.4T 0.034 0.389 0.044 0.980 0.027 0.423 0.043 0.982 τ -FBPW_{, B = 0.5T} _0.036 _0.385 _0.046 _0.981 _0.029 _0.421 _0.043 _0.983 ADF 0.058 0.240 0.051 0.967 0.057 0.242 0.052 0.967 DF-GLS 0.085 0.542 0.060 0.999 0.086 0.480 0.061 0.986 DF-GLS-trend 0.082 0.219 0.054 0.955 0.080 0.206 0.056 0.932 EL 0.106 0.089 0.066 0.573 0.107 0.087 0.066 0.559

Note: Simulation results are reported for 100,000 replications. The zero-trend dt = 0 is considered for all t = 1, . . . , T .

The AR(1) process is given by ut= 0.5ut−1+ t. All innovations are simulated independently as standard normal random

variables. For the small-b and fixed-b tests, the lag order p refers to the pre-whitening scheme, and, for the conventional tests, p is equal to the augmentation order. The random initial condition is simulated from T lagged innovations. For ρ = 1, the rejection frequencies are based on the asymptotic critical values for a significance level of 5%, while, for ρ = 0.9, the values are size-adjusted.

(27)

Table 1.4: Size and size-adjusted powers under different trends and i.i.d. errors (1/2) T = 100, ρ = 1 T = 100, ρ = 0.9 T = 300, ρ = 1 T = 300, ρ = 0.9 λ = 3 λ = 9 λ = 3 λ = 6 λ = 9 λ = 3 λ = 9 λ = 3 λ = 6 λ = 9 sharp break τ -SB, B = T0.5 _0.056 _0.039 _0.248 _0.162 _0.104 _0.056 _0.053 _0.816 _0.725 _0.586 τ -SB, B = T0.6 _0.056 _0.039 _0.281 _0.159 _0.085 _0.058 _0.054 _0.928 _0.834 _0.656 τ -SB, B = T0.7 0.053 0.038 0.296 0.124 0.043 0.056 0.054 0.970 0.856 0.566 τ -SB, B = T0.8 _0.044 _0.048 _0.281 _0.075 _0.014 _0.047 _0.052 _0.969 _0.688 _0.173 τ -FB, B = 0.2T 0.051 0.042 0.293 0.142 0.061 0.051 0.054 0.972 0.845 0.511 τ -FB, B = 0.3T 0.056 0.056 0.284 0.093 0.024 0.053 0.063 0.970 0.720 0.224 τ -FB, B = 0.4T 0.062 0.081 0.280 0.077 0.013 0.054 0.074 0.968 0.666 0.146 τ -FB, B = 0.5T 0.062 0.085 0.292 0.088 0.016 0.053 0.073 0.972 0.690 0.162 ADF 0.050 0.023 0.158 0.027 0.002 0.051 0.040 0.887 0.255 0.006 DF-GLS 0.078 0.065 0.364 0.056 0.003 0.058 0.058 0.983 0.630 0.063 DF-GLS-trend 0.069 0.057 0.238 0.134 0.060 0.053 0.053 0.965 0.793 0.414 EL 0.060 0.044 0.110 0.088 0.071 0.053 0.050 0.710 0.569 0.397 u-shaped break τ -SB, B = T0.5 _0.055 _0.022 _0.213 _0.113 _0.069 _0.057 _0.046 _0.785 _0.615 _0.413 τ -SB, B = T0.6 _0.056 _0.022 _0.231 _0.098 _0.049 _0.058 _0.047 _0.898 _0.694 _0.397 τ -SB, B = T0.7 0.057 0.025 0.209 0.052 0.016 0.055 0.048 0.945 0.630 0.197 τ -SB, B = T0.8 _0.040 _0.018 _0.252 _0.065 _0.017 _0.046 _0.044 _0.941 _0.443 _0.046 τ -FB, B = 0.2T 0.053 0.026 0.232 0.080 0.031 0.050 0.050 0.946 0.587 0.147 τ -FB, B = 0.3T 0.056 0.033 0.231 0.059 0.017 0.051 0.054 0.940 0.442 0.048 τ -FB, B = 0.4T 0.052 0.025 0.252 0.066 0.018 0.050 0.046 0.929 0.418 0.040 τ -FB, B = 0.5T 0.045 0.010 0.243 0.061 0.018 0.047 0.031 0.922 0.377 0.032 ADF 0.046 0.011 0.178 0.056 0.019 0.049 0.030 0.911 0.381 0.045 DF-GLS 0.077 0.049 0.374 0.092 0.021 0.058 0.056 0.985 0.658 0.105 DF-GLS-trend 0.063 0.016 0.126 0.018 0.003 0.050 0.036 0.819 0.133 0.002 EL 0.064 0.053 0.108 0.090 0.076 0.055 0.057 0.703 0.547 0.377 continuous break τ -SB, B = T0.5 _0.049 _0.015 _0.264 _0.188 _0.115 _0.053 _0.037 _0.839 _0.818 _0.782 τ -SB, B = T0.6 _0.049 _0.014 _0.299 _0.185 _0.093 _0.054 _0.036 _0.944 _0.916 _0.853 τ -SB, B = T0.7 0.046 0.012 0.327 0.165 0.063 0.053 0.035 0.981 0.937 0.783 τ -SB, B = T0.8 _0.035 _0.011 _0.355 _0.175 _0.059 _0.043 _0.029 _0.989 _0.919 _0.640 τ -FB, B = 0.2T 0.042 0.011 0.315 0.172 0.074 0.046 0.030 0.983 0.932 0.745 τ -FB, B = 0.3T 0.043 0.012 0.331 0.160 0.055 0.046 0.031 0.988 0.914 0.631 τ -FB, B = 0.4T 0.045 0.015 0.351 0.174 0.058 0.047 0.032 0.991 0.934 0.685 τ -FB, B = 0.5T 0.046 0.016 0.360 0.184 0.062 0.047 0.033 0.992 0.941 0.704 ADF 0.046 0.011 0.149 0.019 0.001 0.047 0.029 0.891 0.273 0.007 DF-GLS 0.064 0.016 0.381 0.064 0.004 0.055 0.035 0.984 0.663 0.088 DF-GLS-trend 0.061 0.022 0.215 0.091 0.026 0.050 0.036 0.956 0.712 0.260 EL 0.059 0.048 0.116 0.111 0.104 0.053 0.049 0.754 0.725 0.678

u-shaped break in intercept

τ -SB, B = T0.5 _0.053 _0.018 _0.209 _0.108 _0.063 _0.056 _0.043 _0.784 _0.607 _0.403 τ -SB, B = T0.6 _0.054 _0.018 _0.221 _0.088 _0.042 _0.057 _0.043 _0.897 _0.680 _0.373 τ -SB, B = T0.7 _0.055 _0.019 _0.194 _0.042 _0.012 _0.054 _0.043 _0.940 _0.592 _0.158 τ -SB, B = T0.8 _0.039 _0.015 _0.238 _0.054 _0.013 _0.045 _0.040 _0.933 _0.381 _0.027 τ -FB, B = 0.2T 0.052 0.021 0.214 0.065 0.025 0.049 0.045 0.941 0.538 0.108 τ -FB, B = 0.3T 0.054 0.026 0.214 0.046 0.012 0.051 0.050 0.931 0.382 0.027 τ -FB, B = 0.4T 0.051 0.020 0.238 0.054 0.012 0.049 0.042 0.923 0.364 0.024 τ -FB, B = 0.5T 0.043 0.008 0.237 0.057 0.014 0.047 0.029 0.919 0.340 0.022 ADF 0.043 0.007 0.118 0.012 0.001 0.047 0.026 0.785 0.075 0.000 DF-GLS 0.075 0.037 0.345 0.067 0.011 0.058 0.049 0.990 0.681 0.098 DF-GLS-trend 0.063 0.016 0.126 0.018 0.003 0.050 0.036 0.819 0.133 0.002 EL 0.064 0.053 0.108 0.090 0.076 0.055 0.057 0.703 0.547 0.377

Note: Simulation results are reported for 100,000 replications. The errors utare simulated independently as standard normal

random variables. The series are not pre-whitened. For ρ = 1, the rejection frequencies are based on the asymptotic critical values for a significance level of 5%, while, for ρ = 0.9, the values are size-adjusted.

(28)

Table 1.5: Size and size-adjusted powers under different trends and i.i.d. errors (2/2) T = 100, ρ = 1 T = 100, ρ = 0.9 T = 300, ρ = 1 T = 300, ρ = 0.9 λ = 3 λ = 9 λ = 3 λ = 6 λ = 9 λ = 3 λ = 9 λ = 3 λ = 6 λ = 9 LSTAR break τ -SB, B = T0.5 _0.052 _0.022 _0.269 _0.211 _0.145 _0.055 _0.043 _0.840 _0.826 _0.800 τ -SB, B = T0.6 _0.051 _0.020 _0.308 _0.209 _0.115 _0.056 _0.042 _0.945 _0.926 _0.883 τ -SB, B = T0.7 0.047 0.017 0.332 0.171 0.063 0.053 0.038 0.983 0.949 0.835 τ -SB, B = T0.8 _0.036 _0.014 _0.348 _0.161 _0.048 _0.044 _0.032 _0.988 _0.907 _0.598 τ -FB, B = 0.2T 0.043 0.016 0.325 0.189 0.086 0.048 0.034 0.984 0.944 0.795 τ -FB, B = 0.3T 0.045 0.016 0.334 0.160 0.053 0.047 0.034 0.987 0.908 0.612 τ -FB, B = 0.4T 0.047 0.018 0.346 0.163 0.048 0.048 0.036 0.988 0.904 0.571 τ -FB, B = 0.5T 0.048 0.020 0.355 0.171 0.051 0.049 0.037 0.989 0.907 0.574 ADF 0.049 0.019 0.179 0.037 0.004 0.050 0.037 0.926 0.416 0.030 DF-GLS 0.070 0.029 0.425 0.104 0.010 0.055 0.042 0.990 0.797 0.215 DF-GLS-trend 0.063 0.033 0.248 0.147 0.068 0.052 0.040 0.972 0.854 0.549 EL 0.059 0.046 0.115 0.112 0.106 0.053 0.050 0.751 0.716 0.666

offsetting LSTAR break

τ -SB, B = T0.5 _0.050 _0.018 _0.267 _0.196 _0.125 _0.053 _0.040 _0.841 _0.822 _0.786 τ -SB, B = T0.6 _0.050 _0.016 _0.297 _0.188 _0.097 _0.055 _0.038 _0.945 _0.919 _0.859 τ -SB, B = T0.7 0.046 0.014 0.324 0.165 0.064 0.053 0.035 0.981 0.935 0.770 τ -SB, B = T0.8 _0.035 _0.011 _0.338 _0.153 _0.053 _0.043 _0.029 _0.985 _0.877 _0.506 τ -FB, B = 0.2T 0.042 0.013 0.319 0.179 0.078 0.046 0.031 0.983 0.928 0.727 τ -FB, B = 0.3T 0.044 0.014 0.334 0.159 0.059 0.047 0.031 0.985 0.882 0.530 τ -FB, B = 0.4T 0.045 0.014 0.336 0.154 0.054 0.047 0.032 0.980 0.832 0.421 τ -FB, B = 0.5T 0.046 0.014 0.326 0.137 0.043 0.047 0.032 0.979 0.798 0.336 ADF 0.052 0.048 0.239 0.121 0.046 0.050 0.044 0.978 0.838 0.476 DF-GLS 0.068 0.023 0.385 0.079 0.008 0.055 0.039 0.969 0.532 0.049 DF-GLS-trend 0.061 0.018 0.191 0.061 0.011 0.049 0.033 0.932 0.532 0.088 EL 0.060 0.050 0.116 0.114 0.109 0.053 0.050 0.755 0.736 0.704 triangular break τ -SB, B = T0.5 _0.051 _0.020 _0.267 _0.204 _0.136 _0.055 _0.042 _0.840 _0.822 _0.793 τ -SB, B = T0.6 _0.050 _0.019 _0.308 _0.205 _0.114 _0.056 _0.041 _0.945 _0.924 _0.879 τ -SB, B = T0.7 0.047 0.016 0.339 0.191 0.083 0.054 0.039 0.983 0.951 0.829 τ -SB, B = T0.8 _0.036 _0.012 _0.346 _0.172 _0.065 _0.045 _0.032 _0.983 _0.871 _0.526 τ -FB, B = 0.2T 0.043 0.015 0.331 0.200 0.097 0.048 0.034 0.984 0.945 0.788 τ -FB, B = 0.3T 0.046 0.016 0.343 0.184 0.075 0.048 0.035 0.983 0.880 0.555 τ -FB, B = 0.4T 0.046 0.016 0.347 0.171 0.065 0.048 0.034 0.977 0.817 0.418 τ -FB, B = 0.5T 0.047 0.016 0.349 0.170 0.063 0.048 0.034 0.982 0.838 0.441 ADF 0.052 0.043 0.231 0.098 0.025 0.051 0.046 0.971 0.768 0.331 DF-GLS 0.069 0.023 0.470 0.179 0.051 0.056 0.039 0.990 0.823 0.326 DF-GLS-trend 0.059 0.018 0.193 0.053 0.009 0.051 0.033 0.919 0.467 0.057 EL 0.060 0.055 0.116 0.113 0.113 0.053 0.050 0.760 0.755 0.746 Fourier break τ -SB, B = T0.5 _0.048 _0.014 _0.258 _0.176 _0.098 _0.053 _0.037 _0.841 _0.820 _0.779 τ -SB, B = T0.6 _0.048 _0.012 _0.284 _0.160 _0.071 _0.054 _0.035 _0.944 _0.914 _0.839 τ -SB, B = T0.7 _0.044 _0.010 _0.301 _0.130 _0.043 _0.052 _0.032 _0.980 _0.914 _0.658 τ -SB, B = T0.8 _0.033 _0.007 _0.310 _0.115 _0.033 _0.042 _0.026 _0.974 _0.743 _0.250 τ -FB, B = 0.2T 0.041 0.009 0.299 0.144 0.054 0.046 0.028 0.981 0.895 0.578 τ -FB, B = 0.3T 0.042 0.009 0.304 0.120 0.037 0.046 0.027 0.974 0.762 0.279 τ -FB, B = 0.4T 0.043 0.009 0.309 0.114 0.033 0.046 0.028 0.963 0.673 0.185 τ -FB, B = 0.5T 0.044 0.009 0.308 0.113 0.032 0.046 0.028 0.968 0.690 0.187 ADF 0.049 0.027 0.203 0.065 0.010 0.049 0.037 0.955 0.623 0.140 DF-GLS 0.064 0.015 0.430 0.140 0.041 0.054 0.034 0.989 0.784 0.235 DF-GLS-trend 0.058 0.011 0.160 0.032 0.004 0.049 0.028 0.891 0.318 0.016 EL 0.061 0.061 0.117 0.117 0.117 0.054 0.054 0.761 0.761 0.761

Note: Simulation results are reported for 100,000 replications. The errors utare simulated independently as standard normal

random variables. The series are not pre-whitened. For ρ = 1, the rejection frequencies are based on the asymptotic critical values for a significance level of 5%, while, for ρ = 0.9, the values are size-adjusted.

(29)

Table 1.6: Size and size-adjusted powers under different trends and AR(1) errors T = 100, ρ = 1 T = 100, ρ = 0.9 T = 300, ρ = 1 T = 300, ρ = 0.9 λ = 3 λ = 9 λ = 3 λ = 9 λ = 3 λ = 9 λ = 3 λ = 9 sharp break τ -SBPW, B = T0.5 _0.004 _0.004 _0.296 _0.150 _0.015 _0.011 _0.797 _0.643 τ -SBPW, B = T0.6 _0.015 _0.013 _0.310 _0.152 _0.031 _0.028 _0.888 _0.759 τ -SBPW, B = T0.7 0.027 0.027 0.323 0.131 0.042 0.041 0.943 0.798 τ -SBPW_{, B = T}0.8 _0.026 _0.038 _0.320 _0.092 _0.040 _0.046 _0.958 _0.674 τ -FBPW_{, B = 0.2T} _0.021 _0.024 _0.320 _0.143 _0.039 _0.043 _0.947 _0.787 τ -FBPW_{, B = 0.3T} _0.035 _0.051 _0.322 _0.109 _0.045 _0.059 _0.956 _0.692 τ -FBPW_{, B = 0.4T} _0.042 _0.075 _0.319 _0.094 _0.047 _0.064 _0.958 _0.657 τ -FBPW, B = 0.5T 0.043 0.075 0.323 0.109 0.048 0.061 0.960 0.672 u-shaped break τ -SBPW_{, B = T}0.5 _0.005 _0.004 _0.304 _0.149 _0.014 _0.010 _0.801 _0.600 τ -SBPW_{, B = T}0.6 _0.015 _0.013 _0.310 _0.134 _0.031 _0.026 _0.886 _0.676 τ -SBPW, B = T0.7 _0.030 _0.033 _0.308 _0.084 _0.042 _0.041 _0.936 _0.646 τ -SBPW, B = T0.8 _0.022 _0.017 _0.331 _0.112 _0.039 _0.038 _0.948 _0.521 τ -FBPW, B = 0.2T 0.023 0.026 0.313 0.116 0.039 0.046 0.939 0.616 τ -FBPW_{, B = 0.3T} _0.032 _0.036 _0.321 _0.099 _0.043 _0.050 _0.947 _0.517 τ -FBPW_{, B = 0.4T} _0.032 _0.025 _0.329 _0.113 _0.043 _0.039 _0.946 _0.504 τ -FBPW_{, B = 0.5T} _0.029 _0.011 _0.326 _0.105 _0.042 _0.031 _0.947 _0.474 continuous break τ -SBPW_{, B = T}0.5 _0.004 _0.003 _0.332 _0.180 _0.015 _0.013 _0.825 _0.737 τ -SBPW_{, B = T}0.6 _0.014 _0.010 _0.340 _0.182 _0.031 _0.029 _0.904 _0.830 τ -SBPW_{, B = T}0.7 _0.025 _0.018 _0.350 _0.170 _0.042 _0.038 _0.952 _0.862 τ -SBPW_{, B = T}0.8 _0.022 _0.016 _0.359 _0.174 _0.038 _0.036 _0.970 _0.847 τ -FBPW, B = 0.2T 0.018 0.013 0.348 0.176 0.038 0.034 0.956 0.858 τ -FBPW, B = 0.3T 0.027 0.020 0.355 0.164 0.042 0.039 0.968 0.841 τ -FBPW, B = 0.4T 0.032 0.025 0.357 0.174 0.044 0.041 0.972 0.859 τ -FBPW_{, B = 0.5T} _0.034 _0.027 _0.357 _0.181 _0.046 _0.042 _0.973 _0.868 LSTAR break τ -SBPW, B = T0.5 0.004 0.002 0.292 0.145 0.015 0.013 0.797 0.669 τ -SBPW_{, B = T}0.6 _0.013 _0.007 _0.305 _0.157 _0.031 _0.028 _0.893 _0.803 τ -SBPW_{, B = T}0.7 _0.024 _0.014 _0.323 _0.142 _0.041 _0.037 _0.948 _0.854 τ -SBPW_{, B = T}0.8 _0.020 _0.012 _0.333 _0.136 _0.039 _0.034 _0.965 _0.816 τ -FBPW_{, B = 0.2T} _0.017 _0.010 _0.320 _0.153 _0.038 _0.033 _0.952 _0.847 τ -FBPW_{, B = 0.3T} _0.026 _0.015 _0.328 _0.136 _0.042 _0.037 _0.964 _0.817 τ -FBPW, B = 0.4T 0.031 0.019 0.331 0.138 0.044 0.040 0.966 0.813 τ -FBPW, B = 0.5T 0.034 0.021 0.331 0.144 0.045 0.041 0.968 0.810 Fourier break τ -SBPW_{, B = T}0.5 _0.004 _0.003 _0.335 _0.217 _0.015 _0.013 _0.826 _0.755 τ -SBPW, B = T0.6 _0.013 _0.009 _0.342 _0.205 _0.032 _0.028 _0.905 _0.839 τ -SBPW, B = T0.7 0.025 0.015 0.349 0.180 0.041 0.036 0.951 0.846 τ -SBPW, B = T0.8 0.021 0.013 0.353 0.171 0.038 0.034 0.961 0.729 τ -FBPW_{, B = 0.2T} _0.018 _0.011 _0.348 _0.192 _0.038 _0.033 _0.954 _0.830 τ -FBPW_{, B = 0.3T} _0.027 _0.017 _0.352 _0.173 _0.042 _0.037 _0.960 _0.738 τ -FBPW_{, B = 0.4T} _0.032 _0.020 _0.349 _0.170 _0.044 _0.039 _0.959 _0.681 τ -FBPW_{, B = 0.5T} _0.034 _0.021 _0.348 _0.167 _0.045 _0.039 _0.962 _0.695

Note: Simulation results are reported for 100,000 replications. The errors ut are simulated from ut = 0.5ut−1+ t with

independent standard normal innovations, and the series are pre-whitened with a lag order that is determined from the BIC. For ρ = 1, the rejection frequencies are based on the asymptotic critical values for a significance level of 5%, while, for ρ = 0.9, the values are size-adjusted.

(30)

Table 1.7: Size and size-adjusted powers of robust tests under constant trend and variance

iid errors, p = 0 AR(1) errors, p is chosen by BIC

T = 100 T = 300 T = 100 T = 300 ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9 ρ = 1 ρ = 0.9 τ -SBH_{, B = T}0.5 _0.063 _0.298 _0.057 _0.849 _0.006 _0.356 _0.016 _0.838 τ -SBH, B = T0.6 _0.064 _0.355 _0.059 _0.953 _0.018 _0.365 _0.033 _0.913 τ -SBH, B = T0.7 _0.062 _0.408 _0.058 _0.988 _0.032 _0.381 _0.044 _0.960 τ -SBH, B = T0.8 0.049 0.434 0.048 0.996 0.032 0.380 0.042 0.976 τ -FBH_{, B = 0.2T} _0.044 _0.348 _0.046 _0.979 _0.020 _0.315 _0.037 _0.942 τ -FBH_{, B = 0.3T} _0.046 _0.386 _0.047 _0.989 _0.028 _0.331 _0.042 _0.960 τ -FBH_{, B = 0.4T} _0.047 _0.400 _0.048 _0.992 _0.033 _0.337 _0.043 _0.966 τ -FBH_{, B = 0.5T} _0.049 _0.402 _0.048 _0.993 _0.034 _0.337 _0.044 _0.970

Note: Simulation results are reported for 100,000 replications. All innovations are simulated independently as standard normal random variables, and the initial condition is x0= 0. The AR(1) process is given by ut= 0.5ut−1+ t. For ρ = 1,

the rejection frequencies are based on the asymptotic critical values for a significance level of 5%, while, for ρ = 0.9, the values are size-adjusted.

The power of the pooled tests depends on the blocklength. In case of no break, a larger blocklength implies higher power results, which is in line with the theoretical findings that those tests have power in a 1/√BT neighborhood of the unit root hypothesis. For blocklengths of B = T0.8 in the small-b case and B = 0.5T in the fixed-b case, the power results are higher than for the ADF test and also larger than those obtained when performing the Dickey-Fuller GLS test under a random initial condition. Hence, we do not lose power under these small-sample specifications (although, asymptotically, those tests have power in a 1/T neighborhood of the unit root hypothesis). Furthermore, smaller sample sizes, such as T0.6 _{in the small-b context and 0.3T in the fixed-b context, still yield} reasonably high power. In particular, the EL test performs much worse in all cases. The size and power results obtained under the AR(1) error specification with both fixed and flexible lag augmentation for the pre-whitening scheme are similar to those produced by i.i.d. errors.

As the tests are designed to yield higher power in the presence of slowly varying trends and breaks, we compare the size-adjusted powers of the tests under the trend specifications presented in Table 1.2 and Figure 1.1. For large break sizes λ, it is shown that the smaller the blocklength, the greater the power results. In most cases, the pooled tests have greater power than the ADF, the DF-GLS, the DF-GLS-trend, and the EL test. Furthermore, the power results of the pooled tests are quite uniform across different trend specifications when compared to those of the conventional tests.

Table 1.6 shows that the pooled tests have reasonable size and power properties under the presence of AR(1) errors and different trend specifications. Furthermore, from Tables 1.7 and 1.8, we can conclude that the heteroskedasticity-robust tests are sized correctly and have good power properties in the presence of a break in the variance and in the trend function.

Three Essays on Structural Stability of Time Series Models